Products: ABAQUS/Standard ABAQUS/Explicit
Analyses performed using acoustic elements, an acoustic medium, and a dynamic procedure can simulate a variety of engineering phenomena including low-amplitude wave phenomena involving fluids such as air and water and “shock” analysis involving higher amplitude waves in fluids interacting with structures.
An acoustic analysis:
is used to model sound propagation, emission, and radiation problems;
can include incident wave loading to model effects such as underwater explosion (UNDEX) on structures interacting with fluids or airborne blast loading on structures;
in ABAQUS/Explicit can include fluid undergoing cavitation when the absolute pressure drops to a limit value;
is performed using one of the dynamic analysis procedures (“Dynamic analysis procedures: overview,” Section 6.3.1);
can be used to model an acoustic medium alone, as in the study of the natural frequencies of vibration of a cavity containing an acoustic fluid;
can be used to model a coupled acoustic-structural system, as in the study of the noise level in a vehicle;
requires the use of acoustic elements and, for coupled acoustic-structural analysis, a surface-based interaction using a tie constraint or, in ABAQUS/Standard, acoustic interface elements;
can be used to obtain the scattered wave solution directly under incident wave loading when the mechanical behavior of the fluid is linear;
can be used to obtain a total wave solution (sum of the incident and the scattered waves) by selecting the total wave formulation, particularly when nonlinear fluid behavior such as cavitation is present in the acoustic medium;
can be used to model problems where the acoustic medium interacts with a structure subjected to large static deformation;
in ABAQUS/Standard can be used with symmetric model generation (“Symmetric model generation,” Section 7.8.1) and symmetric results transfer (“Transferring results from a symmetric mesh or a partial three-dimensional mesh to a full three-dimensional mesh,” Section 7.8.2);
in ABAQUS/Standard can include a coupled structural-acoustic substructure that was previously defined (“Defining substructures,” Section 7.2.2);
can be used to model both interior problems, where a structure surrounds one or more acoustic cavities, and exterior problems, where a structure is located in a fluid medium extending to infinity; and
is applicable only if the effects of shear stress in the acoustic medium are negligible.
is used to model blast effects on structures;
often requires double precision to void roundoff error when ABAQUS/Explicit is used;
may include acoustic elements to model the effects of fluid inertia and compressibility;
may include virtual mass effects to model the effect of an incompressible fluid interacting with a beam structure;
is performed using one of the dynamic analysis procedures (“Dynamic analysis procedures: overview,” Section 6.3.1); and
can be used to model both interior problems, where a structure surrounds one or more fluid cavities, and exterior problems, where a structure is located in a fluid medium extending to infinity.
Acoustic elements model the propagation of acoustic waves and are active only in dynamic analysis procedures. They are most commonly used in the following procedures:
Direct solution, steady-state, harmonic analysis. See “Direct-solution steady-state dynamic analysis,” Section 6.3.4.
Frequency analysis. See “Natural frequency extraction,” Section 6.3.5.
Subspace-based steady-state dynamic analysis. See “Subspace-based steady-state dynamic analysis,” Section 6.3.9.
Explicit dynamic analysis. See “Explicit dynamic analysis,” Section 6.3.3.
Direct time integration analysis. See “Implicit dynamic analysis using direct integration,” Section 6.3.2.
Mode-based transient dynamic analysis. See “Transient modal dynamic analysis,” Section 6.3.7.
Mode-based steady-state dynamic analysis. See “Mode-based steady-state dynamic analysis,” Section 6.3.8.
Dynamic fully coupled temperature-displacement analysis. See “Fully coupled thermal-stress analysis,” Section 6.5.4.
Acoustic elements can be used in a static analysis, but all acoustic effects will be ignored. A typical example is the air cavity in a tire/wheel assembly. In such a simulation the tire is subjected to inflation, rim mounting, and footprint loads prior to the coupled acoustic-structural analysis in which the acoustic response of the air cavity is determined. See “Defining adaptive mesh domains in ABAQUS/Standard,” Section 7.17.6, and “Adaptive meshing and remapping in ABAQUS/Standard,” Section 7.17.7, for more information.
Acoustic elements also can be used in a substructure generation procedure to generate coupled structural-acoustic substructures. Only structural degrees of freedom can be retained. Since coupled substructures will typically only be used in dynamic analyses, the retained eigenmodes will most often be selected as well. In a static analysis involving a substructure containing acoustic elements, the results will differ from the results obtained in an equivalent static analysis without substructures. The reason is that the acoustic-structural coupling is taken into account in the substructure (leading to hydrostatic contributions of the acoustic fluid), while the coupling is ignored in a static analysis without substructures. More details on coupled structural-acoustic substructures can be found in “Defining substructures,” Section 7.2.2.
In ABAQUS/Standard a volumetric drag coefficient, 
, can be defined to simulate fluid velocity-dependent pressure amplitude losses. These occur, for example, when the acoustic medium flows through a porous matrix that causes some resistance (see “Acoustic medium,” Section 12.3.1), such as a sound-deadening material like fiberglass insulation. For direct time integration dynamic analysis we assume there are no significant spatial discontinuities in the quantity 
, where 
is the density of the fluid (acoustic medium), and that the volumetric drag is small at acoustic-structural boundaries. These assumptions, which can limit the applicability of the analysis, are discussed further in “Coupled acoustic-structural medium analysis,” Section 2.9.1 of the ABAQUS Theory Manual.
The direct-solution steady-state dynamic harmonic response procedure is the method of choice for acoustic-structural sound propagation problems, because the gradient of need not be small and because acoustic-structural coupling and damping are not restricted in this formulation. If there is no damping or if damping can be neglected, factoring a real-only matrix can reduce computational time significantly; see “Direct-solution steady-state dynamic analysis,” Section 6.3.4, for details.
Some fluid-solid interaction analyses involve long-duration dynamic effects that more closely resemble structural dynamic analysis than wave propagation; that is, the important dynamics of the structure occur at a time scale that is long compared to the compressional wave speed of the solid medium and the acoustic wave speed of the fluid. Equivalently, in such cases, disturbances of interest in the structure propagate very slowly in comparison to waves in the fluid and compressional waves in the structure. In such instances, modeling of the structure using beams is common. When these structural elements interact with a surrounding fluid, the important fluid effect is due to motions associated with incompressible flow (see “Loading due to an incident dilatational wave field,” Section 6.3.1 of the ABAQUS Theory Manual). These motions result in a perceived inertia added to the structural beam; therefore, this case is usually referred to as the “virtual mass approximation.” For this case ABAQUS allows you to modify the inertia properties of beam elements, as described below. Loads on the structure associated with incident waves in the fluid can be accommodated under this approximation as well.
In a coupled acoustic-structural model the coupled modes are extracted by default in an eigenfrequency extraction step. If coupling is defined in the model, it will be ignored only if either the subspace iteration eigensolver is used or if it is specified that the coupling effects should be ignored for the eigenfrequency extraction step. In either of these cases, acoustic and structural elements may both be present, but the modes are extracted as though the coupling were absent. Thus, when coupling is neglected, the structural elements appear as though the interface with the acoustic elements were free (as though this surface were “in vacuo”), and the acoustic elements behave as though the boundary with the structural elements were rigid.
Since damping is not taken into account in modal extraction, the volumetric drag effect is not considered and any nonreflecting or impedance boundaries behave as though they were rigid. The contributions due to any impedance boundary conditions (element-based or surface-based) or acoustic infinite elements are not included in an eigenfrequency extraction step..
Virtual mass effects defined for beams by adding inertia (“Additional inertia due to immersion in fluid” in “Beam section behavior,” Section 15.3.5) are included in modal analysis: their effect is simply to add inertia to a beam element.
While all the modes extracted in a coupled structural-acoustic analysis are coupled modes, some of them may have predominantly structural contributions while others may have predominantly acoustic contributions. Coupled structural-acoustic eigenmodes can be categorized as follows:
Most generally, an individual mode may exhibit participation in both the fluid and the solid media; this is referred to as a “coupled mode.”
Second, there are the “structural resonance” modes. These are modes corresponding to the eigenmodes of the structure without the presence of the acoustic fluid. The presence of the acoustic fluid has a relatively small effect on the eigenfrequencies and the mode shapes.
Third, there are the “acoustic cavity resonance” modes. These are nonzero eigenfrequency coupled modes that have a significant contribution in the resulting dynamics of the acoustic pressure in mode-based dynamic procedures.
Fourth, if insufficient boundary conditions are specified on the structural part of a model, the frequency extraction procedure will extract rigid body modes. These modes have zero eigenfrequencies (sometimes they appear as either small positive or even negative eigenvalues). However, if sufficient structural degrees of freedom are constrained, these rigid body modes disappear.
Finally, there are the singular acoustic modes, which have zero eigenfrequencies and constant acoustic pressure; they are mathematically analogous to structural rigid body modes. The structural part of the singular acoustic modes corresponds to the quasistatic structural response to constant pressure in unconstrained acoustic regions. These eigenmodes are predominantly acoustic and are important in representing the (low-frequency) acoustic response in mode-based analysis in the presence of acoustic loads, in the same way that rigid body modes are important in the representation of structural motion. As is true for the structural rigid body modes, if a sufficient number of constrained acoustic degrees of freedom is specified (one degree of freedom 8 per acoustic region is enough), the singular acoustic modes will disappear. In models with only one unconstrained acoustic region (the most common case) only one singular acoustic mode will be computed. In general there are as many singular acoustic modes as there are independent unconstrained acoustic regions. If these modes are present, they are always reported first by the Lanczos eigensolver; and a note at the bottom of the eigenfrequency table in the data file provides information about the number of singular acoustic modes.
The generalized masses and effective masses can help distinguish between the various types of modes and can be used to assess which modes are important for subsequent mode-based analyses. In addition, the acoustic contribution to the generalized masses is reported as a fraction for each eigenmode. The closer the value of this fraction is to unity, the more pronounced is the acoustic component of this eigenmode. An acoustic effective mass is also computed for each eigenmode. This scalar quantity is scaled such that when all eigenmodes in a model are extracted, the sum of all acoustic effective masses is equal to 1.0 (minus the contributions from nodes with restrained acoustic degrees of freedom). The acoustic effective mass can be compared between different modes: the higher the acoustic effective mass, the more important (typically) the mode is for accurate representation of the acoustic pressure. For example, the fluid cavity acoustic resonance modes will have larger acoustic effective masses compared to the other modes.
Acoustic medium damping is not considered in most of the procedures that base the response prediction directly on the system's eigenmodes, such as transient modal dynamic analysis or the mode-based steady-state dynamic procedure. These methods should, therefore, not be used for problems with impedance boundary conditions. Modal damping can be used in these procedures (“Material damping,” Section 12.1.1) to model material damping and volumetric drag effects; however, modal damping cannot be used to accurately model the fluid-solid coupling or the impedance boundary effects.
In subspace-based steady-state dynamic analysis, acoustic medium damping and structural material damping are considered, and the acoustic-structural interaction and impedance boundary terms are also included.
If damping is nonexistent or negligible, the mode-based steady-state dynamic procedure is the most computationally efficient alternative to compute the steady-state response. The real-only modal superposition procedures are also effective in acoustics.
By default, the acoustic-structural coupling calculations are based on the original configuration of the fluid domain. However, acoustic elements can also be used in an analysis where the structural domain experiences large deformation prior to the coupled analysis. A typical example is the interior cavity of a tire subjected to structural loads such as inflation, rim mounting, and footprint pressure.
The acoustic elements in ABAQUS do not have displacement degrees of freedom and, therefore, cannot model the deformation of the fluid when the structure undergoes large deformation. ABAQUS/Standard solves the problem of computing the current configuration of the acoustic domain by periodically creating a new acoustic mesh. The new mesh uses the same topology (elements and connectivity) throughout the simulation, but the nodal locations are adjusted so that the acoustic domain conforms to the structural domain on the boundary.
A new acoustic mesh is computed when adaptive meshing is specified and nonlinear geometric effects are considered in any non-perturbation ABAQUS/Standard analysis procedure in which acoustic effects are ignored.
The adaptive meshing features for acoustic analysis are described in detail in “Defining adaptive mesh domains in ABAQUS/Standard,” Section 7.17.6, and “Adaptive meshing and remapping in ABAQUS/Standard,” Section 7.17.7.
In ABAQUS/Standard the initial acoustic static pressure (hydrostatic or ambient) is not modeled by the acoustic formulation and cannot be specified as an initial condition.
In ABAQUS/Explicit the initial acoustic pressure corresponding to the initial static equilibrium (hydrostatic or ambient) can be specified (see “Initial conditions,” Section 19.2.1) and is meaningful only when the acoustic fluid is capable of undergoing cavitation. In problems with possible fluid cavitation the initial acoustic static pressure is taken into account in the cavitation condition; i.e., the sum of the dynamic and static acoustic pressures needs to drop to the cavitation pressure limit for the cavitation to occur. The specified acoustic static pressure is used only in the cavitation condition and does not apply any static loads to the acoustic or structural meshes at their common wetted interface. In addition, the acoustic static pressure is not included in the nodal acoustic pressure degree of freedom.
The initial temperature and field variable values can be specified (“Initial conditions,” Section 19.2.1) for the direct time integration dynamic, explicit dynamic, dynamic fully coupled temperature-displacement, and mode-based transient dynamic analysis procedures. Changes in these variables during the analysis will affect any temperature-dependent or field-variable-dependent acoustic medium properties.
The various boundary conditions that can be applied to an acoustic medium are described below. These include acoustic domain boundaries with stationary rigid walls or symmetry planes, prescribed pressure values such as a free surface with zero dynamic pressure, specified impedance (see “Acoustic loads,” Section 19.4.5), and structural interfaces such as the interface with a ship or a submarine. The radiating (nonreflecting) boundary condition for exterior problems (such as a structure vibrating in an acoustic medium of infinite extent) is implemented as a special case of the impedance boundary condition (see “Acoustic loads,” Section 19.4.5). On any given part of the acoustic domain boundary only one boundary condition type should be applied, except for the combination of the impedance boundary condition and the acoustic-structural interface condition.
The default boundary condition for an acoustic medium is a boundary with a stationary rigid wall or a symmetry plane. The “force” conjugate to pressure in the acoustics formulation in ABAQUS is the normal pressure gradient at the surface divided by the mass density; dimensionally this is equal to a force per unit mass. In the absence of volumetric drag this force per unit mass is equal to the inward acceleration of the acoustic medium. The conjugate variable at a node on the surface is the inward volume acceleration, which is the integral of the inward acceleration of the acoustic medium evaluated over the surface area associated with the node. A “traction-free” surface (one with no boundary conditions, no applied loads, no surface impedance conditions, and no interface elements) is a surface normal to which the acoustic medium undergoes no motion and, thus, corresponds to a rigid, stationary surface adjacent to the fluid. A symmetry plane for the acoustic medium is another “traction-free” surface.
The basic variable in the acoustic medium is pressure (degree of freedom 8). Therefore, this variable can be prescribed at any node in the acoustic model by applying a boundary condition (“Boundary conditions,” Section 19.3.1). Setting the pressure to zero represents a “free surface,” where the pressure does not vary because of the motion of the surface (to account for surface motion effects, see the discussion of impedance below). Prescribing a nonzero value for the pressure represents a sound source.
An amplitude variation can be used to specify the value of the pressure. In a steady-state analysis you can specify both the in-phase (real) part of the pressure (default) and the out-of-phase (imaginary) part of the pressure.
| Input File Usage: | Use either of the following options to define the real (in-phase) part of the boundary condition: |
*BOUNDARY *BOUNDARY, LOAD CASE=1 Use the following option to define the imaginary (out-of-phase) part of the boundary condition: *BOUNDARY, LOAD CASE=2 |
If the acoustic medium is adjacent to a structure, acoustic-structural coupling occurs. The pressure field modeled with acoustic elements creates a normal surface traction on the structure, and the acceleration field modeled with structural elements creates the natural forcing term at the fluid boundary (for details, see “Coupled acoustic-structural medium analysis,” Section 2.9.1 of the ABAQUS Theory Manual).
In ABAQUS/Standard, if the structural and acoustic meshes share nodes at the boundary, lining this boundary with acoustic-structural interface elements (see “Acoustic interface element library,” Section 18.14.2) will enforce the required physical coupling condition. The interface element normals must point into the acoustic medium, which forces continuity of the normal accelerations of the acoustic medium and of the structure at the boundary and ensures that the pressure of the acoustic elements is applied to the structure. Displacements can also be prescribed at such a boundary.
Alternatively, a surface-based procedure can be used to enforce the coupling; in ABAQUS/Explicit the surface-based procedure is the only available method. This method requires that the structural and acoustic meshes use separate nodes. You define surfaces on the structural and fluid meshes and define the interaction between the two meshes using a surface-based tie constraint (see “Mesh tie constraints,” Section 20.3.1). No additional element definitions are required.
The slave surface, the first of the two surfaces specified for the tie constraint, must be element-based; whereas the master surface can be either element- or node-based.
Acoustic infinite elements may form surfaces that can be coupled to structures by using a tie constraint in two different ways. The acoustic infinite element surface may consist of the base (first) facets of the acoustic infinite elements; in this case this surface should be tied to a topologically similar structural surface. The acoustic infinite element edges may also be used to define surfaces ( see “Mesh tie constraints,” Section 20.3.1), which can be tied to solid elements. This approach couples the semi-infinite sides of acoustic infinite elements to solid elements.
| Input File Usage: | Use the following option in an analysis with the fluid mesh surface as the slave: |
*TIE, NAME=fluidslave fluid_surf, struct_surface Use the following option in an analysis with the solid mesh surface as the slave: *TIE, NAME=solidslave struct_surf, fluid_surf |
Although the meshes may be nodally nonconforming at the tied surfaces, mesh refinement affects the accuracy of the coupled solution. In acoustic-solid problems the mesh refinement will depend on the wave speeds in the two media. The mesh for the medium with the lower wave speed should generally be more refined and, therefore, should be the slave surface. If the details of the wave field in the vicinity of the fluid-solid interface are important, the meshes should be of equally high refinement, with the refinement corresponding to the lower wave speed medium. In this case the choice of the master surface is arbitrary. An exception is the case where the acoustic medium must be updated to follow the structure during a large-deformation analysis. In such a case the slave surface must be defined on the acoustic domain. Another exception is the case of fluids coupled to both sides of shell or beam elements (as described below).
In some applications the normal surface traction on the structure created by the acoustic fluid may be negligible compared to other forces in the structural system. For example, a metal motor housing may radiate sound into the surrounding air, but the reaction pressure of the air on the motor may be insignificant to the dynamics of the housing. In these cases the submodeling technique (see “Submodeling,” Section 7.3.1) can be used to solve the system sequentially; that is, the structural analysis (uncoupled from the fluid) is followed by the acoustic analysis (driven by the structure). Usually, this decoupling of the analysis reduces computational cost. The structural system plays the role of the “global” model, and the acoustic fluid is the submodel. The structural displacements on the boundary of the acoustic fluid must be saved to the results file in the global analysis. Since ABAQUS interpolates the fields between the global and submodels, acoustic-structural interface elements can be used. They should be applied to the fluid boundary to be driven by the global structural model.
| Input File Usage: | Global analysis (structural): |
*NSET, NSET=driving_the_fluid *NODE FILE, NSET=driving_the_fluid U Submodel analysis (fluid): *NSET, NSET=fluid_to_be_driven Acoustic interface elements created on NSET=fluid_to_be_driven *SUBMODEL, EXTERIOR TOLERANCE=tolerance fluid_to_be_driven *BOUNDARY, SUBMODEL, STEP=1 fluid_to_be_driven, 1, 3, |
In ABAQUS/Standard there are two alternatives available for modeling a beam (in two dimensions) or shell interacting with fluid on both sides: a surface-based procedure and an element-based procedure. In ABAQUS/Explicit the surface-based procedure must be used.
Use of the surface-based procedure is straightforward. Two surfaces must be defined on the beam or shell: one on the SPOS side and one on the SNEG side. Each surface is then coupled to the fluid using a tie constraint. At least one of the two surfaces on the beam or shell must be a master surface.
In ABAQUS/Standard, if the same nodes are used for the fluid and the beam or shell, acoustic interface elements must be used in the following manner to define acoustic-structural coupling on both sides of a beam or shell element:
Define a second set of nodes coincident with the beam or shell nodes, and constrain the motions of the two sets of nodes together using a PIN-type MPC (“General multi-point constraints,” Section 20.2.2).
Use the first set of nodes to line one side of the beam or shell elements with acoustic interface elements (with the normals of the acoustic interface elements pointing into the fluid).
Use the second set of nodes to line the other side of the beam or shell elements with acoustic interface elements (with the normals pointing into the fluid on the opposite side of the structure, as in Step 2).
The acoustic elements on the first side of the beam or shell elements should be defined using the first set of nodes, and the acoustic elements on the second side of the beam or shell elements should be defined using the second set of nodes.
In ABAQUS virtual mass effects on submerged Timoshenko beam elements can be modeled by specifying additional inertia for the beam. The virtual mass effects are specified as part of the section definition of the beam.
Define the beam section (“Using a beam section integrated during the analysis to define the section behavior,” Section 15.3.6, or “Using a general beam section to define the section behavior,” Section 15.3.7), any additional internal inertia (“Adding inertia to the beam section behavior for Timoshenko beams” in “Beam section behavior,” Section 15.3.5), and the beam material properties.
Define the virtual mass effect (“Additional inertia due to immersion in fluid” in “Beam section behavior,” Section 15.3.5).
If the model is to be loaded using an incident wave (“Incident wave loading due to external sources” in “Acoustic loads,” Section 19.4.5), define a surface or surfaces on the beam elements.
The following types of loading can be prescribed in an acoustic analysis, as described in “Acoustic loads,” Section 19.4.5:
Concentrated pressure-conjugate loading.
An impedance condition that specifies the relationship between the pressure of the acoustic medium and the normal motion at the boundary (either element-based or surface-based). Such a condition is applied, for example, to include the effect of small-amplitude “sloshing” in a gravity field or to include the effect of a compressible, possibly dissipative, lining (such as a carpet) between the acoustic medium and a fixed, rigid wall or a structure. This type of condition can also be applied to edge facets of acoustic infinite elements.
Nonreflecting radiation conditions on acoustic boundaries (either element-based or surface-based). An impedance property can be defined to select the appropriate radiating boundary condition taking the radiating surface shape into consideration.
Incident wave loading such as that caused by an underwater explosion. Since this type of loading is usually applied in conjunction with semi-infinite acoustic regions, two alternative modeling formulations are available in ABAQUS. A total pressure-based formulation is provided when the incident wave loading is applied to the exterior of a semi-infinite acoustic mesh. This formulation must be used to handle the incident wave loading when the acoustic medium is capable of cavitation, rendering the fluid material behavior nonlinear. The default scattered pressure formulation is typically used when cavitation is not part of the fluid mechanical behavior and when the loads are applied to fluid-solid interfaces.
For both formulations, when incident wave loading is applied to a given surface, a mathematical jump occurs between the pressures on both sides of the surface because the side from which the incident pressure arrives is implicitly a region of scattered pressure. This jump is handled automatically when the incident wave load is applied to a surface with a nonreflecting impedance condition and when the incident wave load is applied to a fluid-solid interface. However, if the incident wave load is applied to a surface lying between acoustic finite or infinite elements, the jump will not be modeled properly because pressures are continuous between acoustic elements. For this case, low-mass and low-stiffness membrane, shell, or surface elements should be interposed between the acoustic elements to permit the jump in pressure to exist. See “Acoustic loads,” Section 19.4.5, for several examples of incident wave loading.
The following predefined fields can be specified in an acoustic analysis, as described in “Predefined fields,” Section 19.6.1:
Although temperature is not a degree of freedom in acoustic elements, nodal temperatures can be specified. The specified temperature affects temperature-dependent material properties.
The values of user-defined field variables can be specified. These values affect field-variable-dependent material properties.
Only the acoustic medium material model (“Acoustic medium,” Section 12.3.1) is valid for use in an acoustic analysis. The structure in a coupled acoustic-structural analysis can be modeled using any material model. Since acoustic analyses are always performed using a dynamic procedure, the structure's density (“Density,” Section 9.2.1) should usually be defined.
When the acoustic medium is capable of cavitation and the analysis includes incident wave loading, a total pressure-based formulation must be used. Either the default scattered wave formulation or the total wave formulation can be used if the cavitation is absent or the problem has no incident wave loading.
For beam elements using the virtual mass approximation, the relevant data are specified as part of the beam section definition.
ABAQUS provides a set of elements for modeling an acoustic medium undergoing small pressure changes. In addition, ABAQUS/Standard provides interface elements to couple these acoustic elements to a structural model (see “Choosing the appropriate element for an analysis type,” Section 13.1.3). If interface elements are used, only direct-solution steady-state harmonic (linear) response analysis (“Direct-solution steady-state dynamic analysis,” Section 6.3.4) and transient response analysis (“Implicit dynamic analysis using direct integration,” Section 6.3.2) can be performed.
In ABAQUS/Standard the second-order acoustic elements are generally considerably more accurate than first-order acoustic elements for a given number of degrees of freedom. The acoustic elements in ABAQUS/Explicit are limited to first-order interpolations.
Acoustic elements cannot be used together with hydrostatic fluid elements.
We often need to model an exterior problem, such as a structure vibrating in an acoustic medium of infinite extent. Impedance-type radiation boundary conditions can be used to model the motions of waves out of the mesh. In addition, ABAQUS provides acoustic infinite elements for this class of problems.
In this case acoustic elements are used to model the region between the structure and a simple geometric surface (located away from the structure), and a radiating (nonreflecting) boundary condition is applied at that surface. The radiating boundary conditions are approximate, so that the error in an exterior acoustic analysis is controlled not only by the usual finite element discretization error but also by the error in the approximate radiation condition. In ABAQUS the radiation boundary conditions converge to the exact condition in the limit as they become infinitely distant from the radiating structure. In practice, these radiation conditions provide accurate results when the distance between the surface and the structure is at least one-half of the longest characteristic or responsive structural wavelength.
For details, see “Acoustic loads,” Section 19.4.5.
Acoustic infinite elements are provided for modeling exterior problems (“Infinite elements,” Section 14.2.1). These elements have surface topology: line and quadratic segments in two-dimensional and axisymmetric problems and triangles and quadrilaterals in three-dimensional problems. Generally, the acoustic infinite elements are defined on a terminating surface of a region of acoustic finite elements. The infinite element formulation is considerably more accurate than the impedance-type radiation boundary conditions in cases where the wave field impinging on the terminating surface has many complex features. The radiation boundary conditions are relatively simple, equivalent to a “zeroth-order” infinite element; the acoustic infinite elements implemented in ABAQUS are of variable order, up to ninth.
Acoustic infinite elements can be coupled directly to structural surfaces by using a surface-based tie constraint: this may provide adequate accuracy in some applications. In general cases the acoustic infinite elements are defined on the terminating surface of the acoustic finite element mesh. The diameter of the acoustic finite element mesh can be considerably smaller, for a given solution accuracy, than is the case when using radiation boundary conditions.
The nodal connectivity on the acoustic infinite element defines the element's surface topology. To complete the element formulation, the surface topology must be mapped into the infinite domain. This mapping requires a reference point, given in the element section property definition. The reference point serves to define a characteristic length used in the coordinate mapping. In the ideal case of acoustic radiation from a spherical surface, the correct placement of the reference point is the center of the sphere. In general, the acoustic infinite elements produce the most accurate results when the reference node is located near the center of the region enclosed by the infinite elements.
Nodal normal vectors are required for an accurate mapping of the infinite domain. The nodal normal vectors must point into the infinite domain and are used to define the portion of the infinite domain treated by a particular infinite element. To cover the infinite domain without overlap, each node attached to an infinite element must have a unique normal. The nodal normal vectors are specified or calculated as follows.
User-specified alternative nodal normals (“Normal definitions at nodes,” Section 2.1.4) are ignored for acoustic infinite elements and, therefore, cannot be used to define normal directions for acoustic elements. Over the element's surface topology, the normal vectors must be divergent; that is, the area mapped (in two dimensions) or the volume mapped (in three dimensions) must increase with distance into the infinite domain. To ensure this criteria, the normal vectors at each acoustic infinite element node are defined to lie along the vector between that node and the reference point given in the element section property definition. See “Infinite elements,” Section 14.2.1, for more information.
For reasonable accuracy, at least six representative internodal intervals of the acoustic mesh should fit into the shortest acoustic wavelength present in the analysis. In steady-state analyses the shortest wavelength will occur in the medium with the lowest speed of sound, at the highest frequency analyzed. In transient analyses the shortest wavelength present is more difficult to determine before an analysis: it is reasonable to estimate this wavelength using the highest frequency present in the loads or prescribed boundary conditions. An “internodal interval” is defined as the distance from a node to its nearest neighbor in an element; that is, the element size for a linear element or half of the element size for a quadratic element. At a fixed internodal interval, quadratic elements are more accurate than linear elements. The level of refinement chosen for the acoustic medium should be reflected in the solid medium as well: the solid mesh should be sufficiently refined to accurately model flexural, compressional, and shear waves.
The level of mesh refinement required depends on the application. Any finite element discretization of a domain in which waves propagate introduces a certain amount of error per wavelength. In meshes that are small in terms of wavelengths, relatively coarse (for example, six internodal intervals per wavelength) meshes may be adequate. For meshes that contain many wavelengths at the frequency of interest, the per-wavelength finite element discretization error accumulates, generally necessitating greater levels of refinement. In these larger meshes the accumulated per-wavelength error may be present throughout the mesh if refinement is inadequate.
The acoustic wavelength decreases with increasing frequency, so there is an upper frequency limit for a given mesh. Let 
represent the maximum internodal interval of an element in a mesh, 
the number of internodal intervals we desire per acoustic wavelength (
is recommended), 
the cyclical frequency of excitation, and 
the speed of sound, where 
is the bulk modulus of the acoustic medium and is its density. The requirements are then expressed as
meters per second. Based on a value of 
, the following table gives some values for maximum internodal distances to model given maximum frequencies accurately:| Maximum Frequency of Interest, | Maximum Internodal Interval, |
|---|---|
| 100 Hz | < 430 mm |
| 500 Hz | < 86 mm |
| 1000 Hz | < 43 mm |
| 20 kHz | < 2.1 mm |
For exterior problems the accuracy of an analysis also depends on the accuracy of the absorbing boundary condition. As mentioned above, the absorbing boundary impedance conditions implemented in ABAQUS are used with a standoff thickness 
of acoustic finite elements between the acoustic sources and the radiating boundary. Since the approximate radiation conditions converge to the exact condition in the limit of infinite standoff, a greater standoff thickness improves the accuracy of the solution. The standoff thickness is expressed as 
wavelengths at the minimum frequency to be analyzed:
:Minimum Frequency of Interest,  | Radiation Boundary Standoff, |
|---|---|
| 100 Hz | > 1140 mm |
| 500 Hz | > 230 mm |
| 1000 Hz | > 114 mm |
| 20 kHz | > 5.7 mm |
The computational requirements for an exterior problem thus depend on both the radiation boundary standoff and the internodal distance. The number of nodes 
in a model depends on the volume of the mesh, controlled by and the spatial dimension 
, and the mesh density, controlled by . The exact number of nodes depends on the details of the model, but the expression
, you can control the size of an analysis by splitting the band. For example, if the overall frequency range of interest is 100 to 10000 Hz, a single spherical mesh covering this band in three dimensions has size
and 
, and creating an exterior mesh for each band, results in two analyses of size 
In coupled acoustic-structural systems there usually exist different wave speeds for the fluid and solid media. In the region of the acoustic-structural interface, the wave phenomena in both media may exhibit length scales characteristic of the slower medium; that is, the length scale of the wave dynamics may be as short as the shorter wavelength, corresponding to the lower wave speed. This result follows from the fact that the two media are coupled at the boundary. The region near the acoustic-structural interface where these effects are important is usually no thicker than the shorter wavelength.
For example, in an analysis involving water interacting with rubber, the wave speed in the rubber may be much lower than that of water. A finite element mesh used to model this problem in detail would require refinement down to six (or more) nodes per shorter wavelength, on both sides of the interface. On the water side (faster, longer wavelength) accuracy will probably not be compromised significantly if this region of high refinement extends no further into the water than one short wavelength. Of course, in some analyses the effects in the vicinity of the interface may be unimportant. Then, the two meshes can be refined only so far as to represent their own characteristic wavelengths accurately.
Nodal output variable POR (pressure magnitude at the nodes of the acoustic elements) is available for an acoustic medium (in ABAQUS/CAE this output variable is called PAC). When the scattered wave formulation (default) is used with incident wave loading, output variable POR represents only the scattered pressure response of the model and does not include the incident wave loading itself. When the total wave formulation is used, output variable POR represents the total dynamic acoustic pressure, which includes contributions from both incident and scattered waves as well as the dynamic effects of fluid cavitation. For either formulation output variable POR does not include the acoustic static pressure.
In ABAQUS/Explicit an additional nodal output variable PABS (the absolute pressure, equal to the sum of POR and the acoustic static pressure) is available.
For steady-state dynamic analysis POR is complex and can be displayed in several forms in the Visualization module of ABAQUS/CAE. In this case the phase angle (PPOR) is available as output to the data (.dat) and results (.fil) files.
Several additional secondary quantities are available for multidimensional acoustic finite elements in direct-solution steady-state dynamic or subspace-based steady-state dynamic analysis. The acoustic particle velocity at any material point is
In steady-state dynamic analysis, additional nodal output quantities are available for acoustic infinite elements. PINF denotes the complex pressure coefficients of the infinite element shape functions. These coefficients can be used to visualize the exterior acoustic field (i.e., within the volume of the acoustic infinite elements) using scripting in the Visualization module of ABAQUS/CAE; see “Using infinite elements to compute and view the results of an acoustic far-field analysis,” Section 8.10.12 of the ABAQUS Scripting User's Manual. INFN is the normal vector used by the acoustic infinite element to define the element volume. INFR denotes the radius used for the element at that node, and INFC denotes the element cosine; that is, the minimum dot product between the nodal normal vector and the acoustic infinite element facet normal vectors attached to that node. See “Acoustic infinite elements,” Section 3.3.2 of the ABAQUS Theory Manual, for more complete descriptions of these quantities. INFN, INFR, INFC are useful in debugging a model using acoustic infinite elements; consequently, it is sometimes valuable to perform a steady-state dynamics, direct analysis on a model to visualize this information.
The following is an example of the step definition for a direct-solution steady-state dynamic acoustic analysis that looks for the response of a model at six frequencies ranging linearly from 
to 
cycles/time. The pressure at node set INPUT (nodes at the boundary) is prescribed to have an in-phase component of 3.0 and an out-of-phase component of –4.0 (i.e., a complex value of 
). An in-phase inward volume acceleration of 40.0 is specified at node 10.
On the surface LINER1 an impedance is defined based on the impedance property named CARPET1. On the second face of all of the elements in element set PAD, another surface impedance based on CARPET1 is defined. On the fourth face of all of the elements in element set END, the default plane wave boundary condition is specified.
Printed output of pressure magnitude and phase is requested for node set OUTPUT. Acoustic pressure and displacement (both magnitude and phase) are written to the results file. All output is written once for each of the six excitation frequencies.
*HEADING … *SURFACE, NAME=LINER1 10, S3 *IMPEDANCE PROPERTY, NAME=CARPET1 Data describing impedance properties as a function of frequency ** *STEP *STEADY STATE DYNAMICS, DIRECT 10, 100, 6 *SIMPEDANCE LINER1, CARPET1 ** *IMPEDANCE PAD, I2, CARPET1 END, I4 ** Apply complex pressure at node set INPUT *BOUNDARY, LOAD CASE=1 INPUT, 8, 8, 3. *BOUNDARY, LOAD CASE=2 INPUT, 8, 8, -4. ** Apply an in-phase inward volume acceleration at node 10 *CLOAD 10, 8, 40. ** Output requests *NODE PRINT, NSET=OUTPUT, TOTALS=YES POR, PPOR *NODE FILE U, PU, POR, PPOR *END STEP
The following is a template of the step definition for an ABAQUS/Explicit acoustic analysis. On the surface SURF an impedance is defined based on the impedance property named IPROP. In addition, impedance is defined on elements or element sets.
*HEADING … *ELEMENT, TYPE=AC2D4R … ** *SURFACE, NAME=SURF Data line to define surface *IMPEDANCE PROPERTY, NAME=IPROP Data describing impedance properties ** *STEP *DYNAMIC, EXPLICIT or *DYNAMIC TEMPERATURE-DISPLACEMENT, EXPLICIT Data line to define incrementation *SIMPEDANCE SURF, IPROP ** *IMPEDANCE Data lines to define impedance on elements or element sets *CLOAD Data line to define acoustic loads *FIELD Data line to define field variable values *END STEP
The following template is representative of a coupled acoustic-structural shock problem:
*HEADING … *ELEMENT, TYPE=…, ELSET=ACOUSTIC Data lines to define acoustic elements *ELEMENT, TYPE=…, ELSET=SOLID Data lines to define solid elements *ELEMENT, TYPE=…, ELSET=BEAM Data lines to define beam elements *BEAM SECTION,ELSET=BEAM,MATERIAL=... Data lines to define the beam stiffness section properties *BEAM FLUID INERTIA Data line to define the beam virtual mass property *SURFACE, NAME=IW_LOAD_ACOUSTIC Data lines to define the acoustic surface loaded by the incident wave *SURFACE, NAME=IW_LOAD_SOLID Data lines to define the solid surface loaded by the incident wave *SURFACE, NAME=IW_LOAD_BEAM Data lines to define the beam surface loaded by the incident wave *SURFACE, NAME=TIE_ACOUSTIC Data lines to define the acoustic surface interface with the solid mesh *SURFACE, NAME=TIE_SOLID Data lines to define the solid surface interface with the acoustic mesh *INCIDENT WAVE PROPERTY, NAME=IWPROP, TYPE=SPHERE Data lines to define a spherical incident wave field *INCIDENT WAVE FLUID PROPERTY Data lines to define the fluid properties for the incident wave field *AMPLITUDE, DEFINITION=TABULAR, NAME=PRESSUREVTIME Data lines to define the pressure-time history at the standoff point *TIE, NAME=COUPLING TIE_ACOUSTIC, TIE_SOLID ** Tie the acoustic mesh to the solid mesh *STEP *DYNAMIC or *DYNAMIC, EXPLICIT *INCIDENT WAVE, PRESSURE AMPLITUDE=PRESSUREVTIME, PROPERTY=IWPROP IW_LOAD_ACOUSTIC, {amplitude} ** Load the acoustic surface *INCIDENT WAVE, PRESSURE AMPLITUDE=PRESSUREVTIME, PROPERTY=IWPROP IW_LOAD_SOLID, {amplitude} ** Load the solid surface IW_LOAD_BEAM, {amplitude} ** Load the beam surface *END STEP